Question: Let $z$ be a complex number such that
\[|z - 12| + |z - 5i| = 13.\]Find the smallest possible value of $|z|.$
Solution: By the Triangle Inequality,
\[|z - 12| + |z - 5i| = |z - 12| + |5i - z| \ge |(z - 12) + (5i - z)| = |-12 + 5i| = 13.\]But we are told that $|z - 12| + |z - 5i| = 13.$  The only way that equality can occur is if $z$ lies on the line segment connecting 12 and $5i$ in the complex plane.

[asy]
unitsize(0.4 cm);

pair Z = interp((0,5),(12,0),0.6);
pair P = ((0,0) + reflect((12,0),(0,5))*(0,0))/2;

draw((12,0)--(0,5),red);
draw((-1,0)--(13,0));
draw((0,-1)--(0,6));
draw((0,0)--Z);
draw((0,0)--P);
draw(rightanglemark((0,0),P,(12,0),20));

dot("$12$", (12,0), S);
dot("$5i$", (0,5), W);
dot("$z$", Z, NE);

label("$h$", P/2, SE);
[/asy]

We want to minimize $|z|$.  We see that $|z|$ is minimized when $z$ coincides with the projection of the origin onto the line segment.

The area of the triangle with vertices 0, 12, and $5i$ is
\[\frac{1}{2} \cdot 12 \cdot 5 = 30.\]This area is also
\[\frac{1}{2} \cdot 13 \cdot h = \frac{13h}{2},\]so $h = \boxed{\frac{60}{13}}.$